Local cohomology with support in ideals of maximal minors

Suppose that kk is a field of characteristic zero, XX is an r×sr×s matrix of indeterminates, where r≤sr≤s, and R=k[X]R=k[X] is the polynomial ring over kk in the entries of XX. We study the local cohomology modules HIi(R), where II is the ideal of RR generated by the maximal minors of XX. We identify the indices ii for which these modules vanish, compute HIi(R) at the highest nonvanishing index, i=r(s−r)+1i=r(s−r)+1, and characterize all nonzero ones as submodules of certain indecomposable injective modules. These results are consequences of more general theorems regarding linearly reductive groups acting on local cohomology modules of polynomial rings.